Finding the best setup for experiments is the main concern of Optimal Experimental Design (OED). We focus on the Bayesian experimental design problem of finding the setup that maximizes the Shannon expected information gain. We propose using the stochastic gradient descent and its accelerated counterpart, which employs Nesterov's method, to solve the optimization problem in OED. We improve the stochastic gradient acceleration with a restart technique, as O'Donoghue and Candes [9] originally proposed for deterministic optimization. We combine these optimization methods with three estimators of the objective function: the double-loop Monte Carlo estimator (DLMC), the Monte Carlo estimator using the Laplace approximation for the posterior distribution (MCLA) and the double-loop Monte Carlo estimator with Laplace-based importance sampling (DLMCIS). Using stochastic gradient methods and Laplace-based estimators together allows us to use expensive and complex models, such as those that require solving a partial differential equation (PDE). From a theoretical viewpoint, we derive an explicit formula to compute the stochastic gradient of the Monte Carlo methods including the Laplace approximation (MCLA) and the Laplace-based importance sampling (DLMCIS). Finally, from a computational standpoint, we study four examples: three based on analytical functions and one based on the finite element method solution to a PDE. The last example is an electrical impedance tomography experiment based on the complete electrode model. In these examples, the accelerated stochastic gradient for the MCLA approximation converges to local maxima in fewer model evaluations by up to five orders of magnitude than the gradient descent with DLMC.

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